Investigation of the solvability of boundary value problems for the nonlocal Poisson equation with periodic conditions in circular domains

Abstract

In this paper, boundary value problems with transformed arguments are studied in the unit ball. The transformation of the arguments is specified using the involution type mapping. These mappings participate both in the equation and in the boundary conditions. The equation under consideration is a nonlocal analog of the Poisson equation. Boundary conditions are specified as a relationship between the value of the desired function in the upper hemisphere and the value of the lower hemisphere. These conditions generalize the known periodic conditions for spherical regions. When studying boundary value problems, the properties of involutive mappings are used. The problems under consideration are solved by reducing them to analogues of boundary value problems with periodic conditions for the classical Poisson equation. Using well-known statements for periodic problems for the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Exact conditions for the solvability of the problems under study are found. Spectral questions related to periodic problems are also studied. Eigenfunctions and eigenvalues of these problems are found.